Question: $f(x, y, z) = xy - x^2z + y^2z$ Is $f$ harmonic? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
A scalar field $f$ is harmonic if its Laplacian is zero. The Laplacian of a scalar field $f$ is the sum of each of its second partial derivatives. $\Delta f = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2} + \dfrac{\partial^2 f}{\partial z^2}$ [What does that triangle mean?] Let's find the second partial derivatives of $f$ ! $\begin{aligned} f_{xx} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right] = \dfrac{\partial}{\partial x} \left[ y - 2xz \right] = -2z \\ \\ f_{yy} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial y} \right] = \dfrac{\partial}{\partial y} \left[ x + 2yz \right] = 2z \\ \\ f_{zz} &= \dfrac{\partial}{\partial z} \left[ \dfrac{\partial f}{\partial z} \right] = \dfrac{\partial}{\partial z} \left[ -x^2 + y^2 \right] = 0 \end{aligned}$ The Laplacian: $\Delta f = -2z + 2z + 0 = 0$ Because the Laplacian of $f$ is zero, $f$ is harmonic.